A Tangent Formula Derived from Patterson-Function Arguments. IV. The Solution of Difference Structures Directly from Superstructure Reflections

Abstract

The possibility of solving by direct methods the difference structure δ of a superstructure using only the intensities of the superstructure reflections h is shown. The refinement of the phases Φ of the strong superstructure E values, which are normalized separately from the E values of the main or substructure reflections, is carried out maximizing the function Zδp′ = Σh(Eh − 〈Eh〉)Ch(Φ), where Ch(Φ) is the amplitude of the structure factor of δ3 expressed in terms of Φ. The maximization is performed with a new tangent formula that only differs from that given previously [Rius (1993). Acta Cryst. A49, 406–409] by an extra summation, i.e. the phase information is now derived from quartets and negative quintets instead of triplets and negative quartets. A preliminary test calculation demonstrates the capability of this tangent formula to solve the difference structure of the mineral wermlandite using only the measured superstructure intensities. Although more tests covering a variety of situations are still required to allow for a generalization, this result seems to confirm the viability of determining the internal structure of reconstructed surfaces by interpreting the corresponding three-dimensional difference Patterson function by direct methods. Access to this function is now possible, since, with the advent of intense synchrotron sources, not only in-plane intensity data but also the corresponding diffraction rods can be measured.